Efficient MIR crosstalk reduction based on silicon-on-calcium fluoride platform with Ge/Si strip arrays

Reduction of the crosstalk (CT) between contiguous photonic components is still a big challenge in fabricating high packing density photonic integrated circuits (PICs). Few techniques to accomplish that goal have been offered in recent years but all in the near-IR region. In this paper, we report a design for realizing a highly efficient CT reduction in the MIR regime, for the first time to the best of our knowledge. The reported structure is based on the silicon-on-calcium-fluoride (SOCF) platform with uniform Ge/Si strip arrays. Using Ge strips shows better CT reduction and longer coupling length (Lc) than the conventional Si based devices over a wide bandwidth in the MIR region. The effect of adding a different number of Ge and Si strips with different dimensions between two adjacent Si waveguides on the Lc and hence on the CT is analyzed using both full vectorial finite element method and 3D finite difference time domain method. An increase in the Lc by 4 orders of magnitude and 6.5 times are obtained using Ge and Si strips, respectively, compared to strips-free Si waveguides. Consequently, crosstalk suppression of − 35 dB and − 10 dB for the Ge and Si strips, respectively, is shown. The proposed structure is beneficial for high packing density nanophotonic devices in the MIR regime, such as switches, modulators, splitters, and wavelength division (de)multiplexers, which are important for MIR communication integrated circuits, spectrometers, and sensors.

www.nature.com/scientificreports/ or substrate (e.g., SiO 2 , air). SOI platform enables the building of several ultra-compact and high-performance photonic components employed in PICs 19 . However, the packing density of PICs is still low, which is a significant roadblock in developing large-scale, low-cost, and multilayer hybrid integrated circuits.
Recently, new approaches have been reported to improve the dense integration of PICs. In this regard, plasmonic waveguides 20 , metal-dielectric hybrid structures 21 , and metamaterial-based structures can be used to shrink the footprint of devices 22 . In the design of PICs, the effect of waveguides on each other must be considered. This is due to mode overlapping between neighboring waveguides, that results in some coupling and CT between the waveguides. However, when the optical modes are strongly confined, the overlap and the CT between waveguides are weak and insignificant. As a result, CT is considered an essential factor of the optical waveguides and device packing density. Hence, various crosstalk reduction techniques have been developed in recent years, such as nanophotonic cloaking 23 and waveguide super-lattices 24 . The results indicate that the majority of CT reduction methods have been obtained at telecommunication wavelengths, 1.3 µm, and 1.55 µm. Also, subwavelength silicon strips and gratings have been introduced into the optical waveguide for controlling guided light in the PICs 25,26 . Consequently, compact coupled waveguide devices have emerged in recent years 27 . Khavasi et al. 25 , have added two subwavelength strips between two adjacent waveguides, where all-dielectric metamaterials generated a highly confined mode. Therefore, a noticeable increase in the L c is induced in comparison to the strips-free case 25 . The L c extends up to two orders of magnitude by adding three silicon strips between two neighboring waveguides when compared to the case of without strips. Yu et al. have achieved numerical results at the same wavelength and size of waveguides 28 . Furthermore, Yang et al. have improved the L c by three orders of magnitude greater than that obtained in 28 by introducing three nonuniform Si strips between the two waveguides 29 . It is worth noting that all of the aforementioned studies have worked in the NIR region, namely at λ = 1.55 µm based on the introduction of silicon strips or grating between standard SOI waveguides.
Silicon-on-calcium-fluoride (SOCF) platform is employed instead of standard SOI at the MIR wavelength region because of its high thermo optical coefficient and polarization sensitivity 30 . In addition, the absorption of SiO 2 is quite high at this spectral regime 31 , therefore, CaF 2 with n ≈ 1.4 (in the MIR spectrum) and a transmission window up to ≈ 9 µm is employed as a substrate. It is worth noting that CaF 2 allows high index contrast and high transmission compared to other materials like sapphire (Al 2 O 3 ) (n ≈ 1.7) and silicon nitride (Si 3 N 4 ) (n ≈ 1.9) with transmission windows of up to ≈ 5.5 µm and ≈ 7 µm, respectively 7 . The wavelength region 3-5 µm contains atmospheric transmission windows, which allow silicon photonics to be used in light detection and ranging systems (LIDAR) and MIR communications 6 . In addition, the wavelength of 3.5 µm is used in sensing applications 32 and microring resonators with low-loss propagation 33 .
In this paper, CT reduction approach is proposed in the MIR region based on SOCF platform, for the first time to the best of our knowledge. The reported structure consists of Si/Ge strips between adjacent waveguides to significantly suppress the crosstalk and increase the coupling length. Firstly, the effect of the geometrical parameters of SOCF based waveguides in strips-free case is carried out. Secondly, comprehensive studies are performed on the effect of adding Si and Ge subwavelength strips with various numbers and dimensions between SOCF based waveguides on the L c and the CT. The full vectorial finite element method (FVFEM) 34-37 via COM-SOL Multiphysics software package (https:// www. comsol. com) is utilized to perform the modal analysis. The 3D finite difference time domain (FDTD) method 38 via Lumerical FDTD software package (https:// www. lumer ical. com) is used to investigate the light propagation through the proposed structure. The simulation results reveal that at λ = 3.5 µm, a very long L c of 3.3 m with a low CT of − 34.65 dB is obtained using 2-Ge strips between two adjacent silicon waveguides. Furthermore, L c of 5.32 mm with CT of − 9.83 dB is achieved in the case of 4-Si strips. The obtained CT (− 34.6 dB) is better than that reported in 39 (− 22.38 dB), where two asymmetric Si-strips between two Si waveguides at λ = 1.55 µm were implemented. In addition, it is even better than that reported in 40 (− 27.71 dB) where three Ge strips have been inserted between two Si waveguides at λ = 1.55 µm.

Waveguide design and principle of operation
The cross-sectional and 3D views of the proposed device based on SOCF platform are depicted in Figs. 1a and b, respectively. This system incorporates two subwavelength Ge or Si strips lying between two parallel silicon ridge waveguides formed on CaF 2 substrate with air cladding. The refractive indices of Ge, Si, and CaF 2 are obtained using Sellmeier equations reported in [41][42][43] . The structural parameters w, d s , d a , are the widths of the Si core, Ge or Si strips, and air slot, respectively. Moreover, d is the edge-to-edge distance between silicon ridge waveguides and h is the height of waveguides and strips.
The proposed system fabrication process flow is depicted in Fig. 1c. Using molecular beam epitaxy, Si layers can be grown epitaxially on a CaF 2 substrate in ultrahigh vacuum 44 where CaF 2 crystal substrates are commercially on hand 45 . Using HBr/Cl 2 chemistry, dry etching of Si can be performed 46 . It is worth mentioning that the same way can be employed to add Si and Ge array strips between the two adjacent waveguides 44 . Further, in the case of Ge strips, wafer bonding of Ge film on insulator can be employed 47 .
The CT reduction between adjacent waveguides depends on decreasing the leakage from one waveguide to its surroundings, thus increasing the L c . The L c is defined as the distance at which the maximum optical power is transferred from one waveguide to the other. According to coupled mode theory, L c is given by 48 : where n s and n a are the real parts of the effective indices of the symmetric and anti-symmetric modes supported by the dual-core structure, respectively.
(1) www.nature.com/scientificreports/ The coupling between any two adjacent waveguides can be controlled by modifying the evanescent wave penetration to the surrounding media and hence controlling the CT 23 . The evanescent wave decay constant in the second medium can be expressed as follows: where, k || x , k | y are the parallel and perpendicular components of the wave vector to the interface between the core and the surrounding medium. ε x , ε y are the dielectric constants of the second medium parallel and perpendicular to the interface, and k 0 is the wave vector in free space. From Eq. (2), the ratio of the permittivity components governs the depth of penetration of the evanescent field into the second medium. So, the decay rate of the evanescent wave for a waveguide can be managed by altering the ratio of permittivity parallel to the interface (ε x ) to that perpendicular to it (ε y ). As a result, penetration depth of the evanescent waves, the L c and hence CT between two adjacent waveguides can be controlled. This can be achieved by altering the permittivity along the horizontal and the vertical directions through changing the dimensions of the strip array.

Numerical results and discussion
Unlike SOI, which has a standard width of 500 nm and height of 220 nm, SOCF waveguides don't have standard dimensions, especially at the MIR region. Accordingly, the geometrical parameters are studied to control the operating wavelength over the MIR regime. By changing the dimensions of the Si waveguides, the real parts of n s and n a modes can be controlled. Therefore, L c changes according to Eq. (1). The modal analysis is performed using FVFEM. The computational domain is divided into triangular elements with a maximum element size of 3 × 10 −8 . Scattering boundary conditions are utilized in all transverse directions to truncate the simulation region. Figure 2 depicts the variation of L c with waveguide dimensions w and h and d. It may be seen in Fig. 2a that L c increases by increasing the Si waveguide's width and height. Additionally, L c increases with increasing d as may be seen in Fig. 2b. However, the main goal of this work is to increase the packing density of nanophotonic devices, i.e., increase the L c with the minimum allowable waveguide's separation. As a result, the dimensions of the SOCF-based waveguides are carefully selected to support the propagation of the strongly confined fundamental transverse electric (TE) mode as shown in the inset of Fig. 2b and obtain a minimum L c at λ = 3.5 µm. Accordingly, the geometrical parameters of Si waveguides are taken as w = 1.5 µm, h = 0.6 µm, and d = 0.5 µm. The calculated L c at these parameters is 844 µm.
Next, uniform strips of Ge or Si are added between the two adjacent waveguides to increase the L c and reduce CT. In this investigation, the effect of the number and width of the inserted strips on the L c and hence on the CT is studied in detail. As the type, number, and dimensions of strips change, the permittivity ratio of modes changes. By controlling this ratio, the penetration depth of the evanescent wave into the surrounding media can be managed. Consequently, the obtained CT can be controlled. Moreover, this change affects the refractive indices n eff of the resultant modes. Hence, the difference in n eff can be minimized to maximize L c according to Eq. (1) and get the lowest CT value.
First, the impact of adding Si strip(s) on the L c is studied and the results are shown in Fig. 3 and Table 1. Figure 3a shows that L c increases by using Si-strips by a factor of 2.5 to 6.5 with respect to the case of without Si strips. Figures 3b-e depict the Electric field (E y ) distribution (2D surface plot) of symmetric and antisymmetric TE modes at effective widths that give maximum L c in each case. By increasing the number of Si strips array, L c increases, and field profiles become more confined in the core region. However, by increasing the number of strips, each individual strip becomes less thick, as summarized in Table 1. It is worth noting, that reducing the thickness of strips will be a challenge in device fabrication. However, less than 50 nm gap between Si structures in Si Bragg grating waveguides have been demonstrated 49 .   www.nature.com/scientificreports/ Figure 4 illustrates the wavelength dependent L c for different studied structures at the optimum value of d s , as summarized in Table 1. As seen from Fig. 4, L c increases for all structures after adding thin Si strips between the two Si waveguides with respect to the case of no strips. Considering the case of no Si strips, the region between the two waveguides is uniform (air) with constant permittivity equal to 1. Thus, ε y is very small, leading to a smaller decay rate k | y in the normal direction according to Eq. (2), which enhanced the coupling process between the two Si waveguides leading to small L c throughout the whole wavelength range. However, by adding Si strips between the two waveguides, ε y becomes large, then the decay rate k | y increases, and the penetration depth of the evanescent wave into the surrounding decrease. Hence the coupling between the two waveguides becomes weak, L c increases. Under certain conditions, the optimum permittivity ratio is attained, L c reaches a maximum value, and a peak appears in the L c curve, as seen in Fig. 4. Additionally, a maximum L c of 8.5 m is obtained in the case of 2-Si strips with d s of 105 nm at λ = 2.5 µm. In the 3 and 4-Si strips' structures, L c has also large values of 0.25 m and 0.3 m, respectively, at λ ≃ 3 µm.
In a similar way, variation of the L c with the number and width of the strips is studied for Ge strips instead of Si strips with the same dimensions of Si waveguides at λ = 3.5 µm, and the obtained results are depicted in Fig. 5a. It is evident from this figure that there is a significant increase in L c for all cases of adding Ge strip(s) between the two adjacent Si ridge waveguides relative to no used strips. Figures 5b-e depict the Electric field (E y ) distribution (2D surface plot) for the analyzed structures with effective width that provides maximum L c . In the case of a single Ge strip, L c becomes ≈ 4 times larger than that in the case of no strips. While for other cases (2, 3, and 4-Ge strips), the L c increases by 4 orders of magnitude, which is considered a very high increase  www.nature.com/scientificreports/ as illustrated in Fig. 5a and Table 2. This large increase in L c is due to the small difference in n eff between the symmetric and antisymmetric modes at the optimum width. The suggested device with Ge strips provides a much wider range of permittivity ratio. For the designs with 2, 3, and 4 Ge strips, the L c curves have two peaks as may be seen in Fig. 5a. Because of reaching close to the optimal permittivity ratio at the effective width of Ge strips. The L c reaches the maximum value of 3.3 m when adding 2-Ge strips between the two Si waveguides with optimum width of 147 nm. In this case, the difference in n eff is very small about 1 × 10 −6 , and hence the L c is extremely large (Eq. 1), allowing waveguides to be placed close to each other with a negligible CT. For more explanation, Table 3 shows the Electric field (E y ) distribution (2D surface plot) and Electric field strength (1D line plot) of symmetric and antisymmetric TE modes at different widths (d s ) of the structure of 2-Ge strips. The utilized geometrical parameters are w = 1.5 µm, h = 0.6 µm, and d = 0.5 µm. As shown in Table 3, when d s of the 2 uniform Ge-strips changes, L c and the field strength also change. The coupling between any two adjacent waveguides can be controlled by altering the permittivity along the horizontal and vertical directions by changing the strips' type (material itself), number, and dimensions. At optimal conditions, the coupling between two waveguides becomes weak, and the coupling length will be very large. This explains why in Fig. 5a the L c curve corresponding to the case of 2-Ge strips has two peaks, at d s = 147 nm and 102 nm. In addition, this interprets why the field strength at one core is high (i.e., E-field is well confined), and the other is weak at 102 nm and especially at 147 nm (Fig. 5c) compared to other studied widths as illustrated in Table 3. However, in the cases of non-optimal strip widths (all rows of Table 3 except the 2nd and 4th rows), the field is divided between the two cores showing a strong coupling between them. This result is revealed by the L c s depicted in the right column of Table 3. An important notice is that with Ge strips of ds = 147 nm, the aspect ratio of the design becomes ≈ 1:4, which is feasible via standard nanofabrication methods. However, the L c decreases with a relative increase in the width of Ge strips (e.g., ≥ 190 nm for 1-Ge Strip) as the air slots between strips and waveguides becomes very thin, which in turn increases the evanescent wave overlapping. As seen from Figs. 3b-e and 5b-e, inserting Ge strips between the two Si cores instead of Si strips makes the field more confined in one core than the other. Thus, the coupling between the two Si waveguides becomes very weak by using Ge strips, and hence L c becomes very large. To further illustrate this idea, Fig. 6. Shows the Electric field (E y ) distribution (2D surface plot) and Electric field strength (1D line plot) of symmetric and antisymmetric TE modes for the structures of 2-Ge strips, 2-Si strips, and that of no strips. It is worth noting that this study is performed at λ = 3.5 µm where the geometrical parameters w, h, d, and d s are taken as 1.5 µm, 0.6 µm, 0.5 µm, 147 nm, respectively. Figure 6. indicates that in the structure with 2-Ge strips the field strength at one core is high (i.e., well confined) while it is weak at the other core relative to the structure with 2-Si Strips. This means that strong coupling between the two waveguides occurs in the case of 2-Si Strips (d s = 147 nm) with L c = 2.47 × 10 3 μm when compared to the case of 2-Ge strips (d s = 147 nm) with L c = 3.3 × 10 6 μm in which the coupling becomes very weak. This is due to the fact that Ge has a higher refractive index (n = 4.035 41 ) than Si (n = 3.4284 42 at λ = 3.5 µm). So, in combination with CaF 2 (n = 1.41 43 ), it has a large index contrast (≈ 2.62). Furthermore, Ge has a wide range of transparency in the MIR region 12 . Therefore, adding Ge strips reduces the field penetration of optical modes from the first waveguide to the second one and hence decreases the CT level 18 .
Coupling length analysis is then done over a wide range of wavelengths for all cases with and without Ge strips at the optimum d s value that gives maximum L c in each case, as shown in Fig. 7. This figure reveals that in cases of 2, 3, and 4-Ge strips, the maximum L c is obtained exactly at λ = 3.5 µm and decreases with moving away from this wavelength. In addition, in the case of structures with zero and 1 strip, L c is below 10 cm throughout the studied wavelength range.
It is worth mentioning that by increasing the L c , the CT decreases. CT (in dB scale) can be calculated via Eq. (3): where P out is the output power at the end of the second waveguide while P in is the input power at the beginning of the first waveguide.
The propagation of the fundamental TE mode is studied at a device length of L = 1000 µm where the geometrical parameters are taken as w = 1.5 µm, h = 0.6 µm, d = 0.5 µm and λ = 3.5 µm for different cases, without adding strips and with adding Si or Ge strips as may be seen in Figs. 8, 9, and 10, respectively. The FDTD method is used to simulate the field propagation through the 3D structure via Lumerical software package. The structure (3) Crosstalk(CT) = 10 log 10 P out P in www.nature.com/scientificreports/ is discretized into very small rectangles with auto non-uniform mesh accuracy 5 and simulation time 15,000 fs to ensure high simulation resolution. The perfectly matched layer (PML) as an absorbing boundary condition is applied to all transverse directions to truncate the simulation domain. It is evident from Fig. 8 that a complete coupling occurs using no strips between the two Si waveguides at L c ≈ 840 µm, that agrees well with the result in Figs. 3 and Table 1. However, adding Si strips affects the coupling strength, especially with increasing the number of Si strips, as indicated in Fig. 9a and b. Figures 10 show that inserting Ge strips significantly decrease the coupling process. As illustrated in Fig. 10b, in the structure of 2-Ge strips, light can travel for more than 1000 µm with a very small CT. That is completely compatible with the results shown in Figs. 5 and Table 2 concerning the structure of 2-Ge strips which indicates that a distance L c of 3.3 m is required for complete coupling between the two waveguides with 2-Ge strips between them. The values of CT are calculated via Eq. (3), as indicated in Table 4, where L ≈ 840 µm and the geometrical parameters are w = 1.5 µm, h = 0.6 µm, d = 0.5 µm, and λ = 3.5 µm. Further, different cases, without and with adding Si and Ge strips between two Si waveguides, are considered, and the obtained CT values are summarized in Table 4. It is worth noting that L is fixed to 840 µm as a required length for complete coupling with no strips between the two adjacent waveguides. In addition, adding Ge strips is more efficient in decreasing CT than Si strips. In this context, adding 2-Ge strips decreases CT to -34.6 dB. This value of CT can be neglected, as mentioned in 50 , where photonic integration density has been evaluated by using Y-branch and optical filters. It has been reported in 50 that if the CT between two neighboring waveguides is less than 30 dB, it can be ignored. Here, the obtained value of CT (− 34.6 dB) is better than that obtained in 39 (− 22.38 dB), where two asymmetric Si-strips between pair of Si waveguides of width 500 nm and half-wavelength center-to-center separation (λ = 1.55 µm) has been considered. In addition, the obtained CT value in this work is better than that reported in 40 (− 27.71 dB), where 3-Ge strips of w = 45 nm and h = 171 nm have been inserted between two Si waveguides with 500 nm width and separated by 250 nm at λ = 1.55 µm.
Based on these results, the longer L c between Si waveguides the lower CT value. This means that the CT between waveguides is basically dependent on the coupling length. Table 5 outlines a comparison between the coupling lengths, basic platform and operating wavelength of the previously reported waveguide structures and the proposed structures. In this work, there are two reported designs based on SOCF platform instead of Table 3. Electric field (E y ) distribution (2D surface plot) and Electric field strength (1D line plot) of symmetric and antisymmetric TE modes at different widths (ds) of the structure of 2-Ge strips between two adjacent silicon waveguides with geometrical parameters w = 1.5 µm, h = 0.6 µm, and d = 0.5 µm. The images of Electric field distribution and field strength are created by COMSOL Multiphysics 5.3, https:// www. comsol. com (license number-17074294) released to Zewail City of Science and Technology, Giza, Egypt. www.nature.com/scientificreports/ standard SOI. The first design is constructed by inserting Si strips between two Si waveguides in which the case of 4-Si strips gives larger L c (5.32 mm) than those reported in 25,51 . However, in 28 , the case of 3-Si strips in the NIR achieves a larger L c of 332 mm. It is worth mentioning that the design based on SOCF platform with 3-Si strips of dimensions w = 500 nm, h = 220 nm, and d = 500 nm (that fit the NIR regime) offers L c of 660 mm at λ = 1.55 μm. The second design depends on inserting Ge strips between two Si waveguides in which the case of 2-Ge strips gives the highest value of L c (3.3 m) that is greater than that reported in 40 . However, in 29 , using 3 nonuniform-Si strips in the NIR has a large L c of 312.2 m. In contrast, the method of adding uniform strips is most straightforward and simple in fabrication. Hence, the reported structure with 2-Ge strips can be utilized as a promising candidate to minimize the CT between any two nearby photonic waveguides in MIR wavelength regime.   www.nature.com/scientificreports/

Conclusion
In this paper, two designs based on the silicon-on-calcium-fluoride platform with uniform Ge/Si strip arrays are reported and analyzed to minimize the CT between two adjacent waveguides in the MIR wavelength range. The coupling length between the two fundamental modes in the two neighboring waveguides increases by increasing the decay rate of the evanescent field from the excited core. This is achieved by controlling the added strips' material, number, and dimensions. The calculated L c increases by factors of 2.5 to 6.5 in the case of inserting Si strips between the two adjacent Si waveguides with respect to the case of no strips achieving allow CT of − 9.83 dB. While L c increases by 4 orders of magnitude by adding Ge strips. The maximum value of L c = 3.3 m is obtained by adding 2-Ge strips with a very small CT of − 34.64 dB. Thus, there is almost no coupling between the two adjacent Si waveguides. To summarize, the proposed approach is beneficial for realizing a highly efficient and broadband CT reduction in the MIR regime. As a consequence, ultra-high dense PICs can be achieved in this spectral range and, therefore will pave the way for further development in different applications such as polarization splitters 52 , polarization rotators 53 , and integrated photonic switches 54 .   www.nature.com/scientificreports/

Data availability
The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.